Page 312 - DMTH401_REAL ANALYSIS

Real Analysis

Notes m*(A) > m*(A  E) + m*(A  E ).
c
The family of all measurable sets is denoted by M. We will see later M is a -algebra and
translation-invariant containing all intervals. The set function m: M  [0, ¥] defined by

m(E) = m*(E) for all E  M
is called Lebesgue measure.
Observe that
c
 E  M  E  M.
   M and   M because m*(A) = m*(A  ) + m*(A  ) for all A  .
c
c
 m*(E) = 0  E M because m*(A  E) + m*(A  E ) = m*(A  E )  m*(A) for all A  .
Proposition: If E , E  M then E  E  M. (Therefore, M is an algebra.)
1 2 1 2
Proof: For all A  R one has
c
m*(A) = m*(A  E ) +m*(A  E ) ( E  M)
1 1 1
c
c
c
= m*(A  E ) + m*(A  E  E ) +m*(A  E  E ) ( E  M)
1 1 2 1 2 2
= m*(A  (E  E )) + m*(A  (E  E ) )
c
1 2 1 2
because m* is subadditive and
A  (E  E ) = (A  E )  (A  E  E ).
c
1 2 1 1 2
Notes Above proposition can be easily extended to a finite union of measurable sets, in
fact it can be extended to a countable union. In order to do so, we need the following result.
Lemma 1: Let E ,E ,...,E be disjoint measurable sets. Then for all A  R, we have
1 2 n
æ é n ö ù n
m* A   E i ÷ = å m* (A  E ).
ç
è ê i 1 úø û i 1 i
= ë
=
Proof: Since E  M, we have
n
n
n
æ
æ
ù
ö ù
é
é
é
æ
m* A   E i ÷ = m* A   n E  E n ÷ ö + m* A   E ù  E c ö
ç
ç
n ÷
ç
è ê i 1 úø û è ê i 1 i ú û ø è ê i 1 i ú û ø
= ë
= ë
= ë
æ
n
é
= m* (A  E ) + m* A   E i ÷ ö ù
ç
n ê
è i 1 úø û
= ë
Repeat the process again and again, until we get
æ é n ö ù n
m* A   E i ÷ = å m* (A  E ).
ç
è ê i 1 úø û i 1 i
= ë
=
Notes If {E } is a sequence of disjoint measurable sets, then
i i
æ
¥
¥
é
ö ù
m* A   E i ÷ = å m* (A  E ).
ç
è ê i 1 úø û i 1 i
= ë
=
This is because for all n   one has
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